Integrand size = 96, antiderivative size = 25 \[ \int \frac {e^{\frac {-3 x^3+\left (x-4 x^2\right ) \log (3)}{3 x+4 \log (3)}} \left (90 x^2-180 x^4+\left (240 x-480 x^3\right ) \log (3)+\left (160+40 x-320 x^2\right ) \log ^2(3)\right )}{9 e^2 x^2+24 e^2 x \log (3)+16 e^2 \log ^2(3)} \, dx=10 e^{-2-x^2+\frac {x}{4+\frac {3 x}{\log (3)}}} x \]
\[ \int \frac {e^{\frac {-3 x^3+\left (x-4 x^2\right ) \log (3)}{3 x+4 \log (3)}} \left (90 x^2-180 x^4+\left (240 x-480 x^3\right ) \log (3)+\left (160+40 x-320 x^2\right ) \log ^2(3)\right )}{9 e^2 x^2+24 e^2 x \log (3)+16 e^2 \log ^2(3)} \, dx=\int \frac {e^{\frac {-3 x^3+\left (x-4 x^2\right ) \log (3)}{3 x+4 \log (3)}} \left (90 x^2-180 x^4+\left (240 x-480 x^3\right ) \log (3)+\left (160+40 x-320 x^2\right ) \log ^2(3)\right )}{9 e^2 x^2+24 e^2 x \log (3)+16 e^2 \log ^2(3)} \, dx \]
Integrate[(E^((-3*x^3 + (x - 4*x^2)*Log[3])/(3*x + 4*Log[3]))*(90*x^2 - 18 0*x^4 + (240*x - 480*x^3)*Log[3] + (160 + 40*x - 320*x^2)*Log[3]^2))/(9*E^ 2*x^2 + 24*E^2*x*Log[3] + 16*E^2*Log[3]^2),x]
Integrate[(E^((-3*x^3 + (x - 4*x^2)*Log[3])/(3*x + 4*Log[3]))*(90*x^2 - 18 0*x^4 + (240*x - 480*x^3)*Log[3] + (160 + 40*x - 320*x^2)*Log[3]^2))/(9*E^ 2*x^2 + 24*E^2*x*Log[3] + 16*E^2*Log[3]^2), x]
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {\left (-180 x^4+\left (240 x-480 x^3\right ) \log (3)+90 x^2+\left (-320 x^2+40 x+160\right ) \log ^2(3)\right ) \exp \left (\frac {\left (x-4 x^2\right ) \log (3)-3 x^3}{3 x+4 \log (3)}\right )}{9 e^2 x^2+24 e^2 x \log (3)+16 e^2 \log ^2(3)} \, dx\) |
\(\Big \downarrow \) 2007 |
\(\displaystyle \int \frac {\left (-180 x^4+\left (240 x-480 x^3\right ) \log (3)+90 x^2+\left (-320 x^2+40 x+160\right ) \log ^2(3)\right ) \exp \left (\frac {\left (x-4 x^2\right ) \log (3)-3 x^3}{3 x+4 \log (3)}\right )}{(3 e x+4 e \log (3))^2}dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {\left (-180 x^4-480 x^3 \log (3)+10 x^2 \left (9-32 \log ^2(3)\right )+40 x \log (3) (6+\log (3))+160 \log ^2(3)\right ) \exp \left (\frac {x \left (-3 x^2-4 x \log (3)+\log (3)\right )}{3 x+4 \log (3)}\right )}{(3 e x+4 e \log (3))^2}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (-\frac {160 \log ^3(3) \exp \left (\frac {x \left (-3 x^2-4 x \log (3)+\log (3)\right )}{3 x+4 \log (3)}-2\right )}{3 (3 x+\log (81))^2}+\frac {40 \log ^2(3) \exp \left (\frac {x \left (-3 x^2-4 x \log (3)+\log (3)\right )}{3 x+4 \log (3)}-2\right )}{3 (3 x+\log (81))}-20 x^2 \exp \left (\frac {x \left (-3 x^2-4 x \log (3)+\log (3)\right )}{3 x+4 \log (3)}-2\right )+10 \exp \left (\frac {x \left (-3 x^2-4 x \log (3)+\log (3)\right )}{3 x+4 \log (3)}-2\right )\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {160}{3} \log ^3(3) \int \frac {\exp \left (\frac {x \left (-3 x^2-4 \log (3) x+\log (3)\right )}{3 x+4 \log (3)}-2\right )}{(3 x+\log (81))^2}dx+\frac {40}{3} \log ^2(3) \int \frac {\exp \left (\frac {x \left (-3 x^2-4 \log (3) x+\log (3)\right )}{3 x+4 \log (3)}-2\right )}{3 x+\log (81)}dx+10 \int \exp \left (\frac {x \left (-3 x^2-4 \log (3) x+\log (3)\right )}{3 x+4 \log (3)}-2\right )dx-20 \int \exp \left (\frac {x \left (-3 x^2-4 \log (3) x+\log (3)\right )}{3 x+4 \log (3)}-2\right ) x^2dx\) |
Int[(E^((-3*x^3 + (x - 4*x^2)*Log[3])/(3*x + 4*Log[3]))*(90*x^2 - 180*x^4 + (240*x - 480*x^3)*Log[3] + (160 + 40*x - 320*x^2)*Log[3]^2))/(9*E^2*x^2 + 24*E^2*x*Log[3] + 16*E^2*Log[3]^2),x]
3.26.36.3.1 Defintions of rubi rules used
Int[(u_.)*(Px_)^(p_), x_Symbol] :> With[{a = Rt[Coeff[Px, x, 0], Expon[Px, x]], b = Rt[Coeff[Px, x, Expon[Px, x]], Expon[Px, x]]}, Int[u*(a + b*x)^(Ex pon[Px, x]*p), x] /; EqQ[Px, (a + b*x)^Expon[Px, x]]] /; IntegerQ[p] && Pol yQ[Px, x] && GtQ[Expon[Px, x], 1] && NeQ[Coeff[Px, x, 0], 0]
Time = 0.72 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.44
method | result | size |
parallelrisch | \(10 \,{\mathrm e}^{-2} x \,{\mathrm e}^{\frac {\left (-4 x^{2}+x \right ) \ln \left (3\right )-3 x^{3}}{4 \ln \left (3\right )+3 x}}\) | \(36\) |
gosper | \(10 \,{\mathrm e}^{-2} x \,{\mathrm e}^{-\frac {x \left (4 x \ln \left (3\right )+3 x^{2}-\ln \left (3\right )\right )}{4 \ln \left (3\right )+3 x}}\) | \(37\) |
risch | \(10 x \,{\mathrm e}^{-\frac {4 x^{2} \ln \left (3\right )+3 x^{3}-x \ln \left (3\right )+8 \ln \left (3\right )+6 x}{4 \ln \left (3\right )+3 x}}\) | \(42\) |
norman | \(\frac {30 x^{2} {\mathrm e}^{-2} {\mathrm e}^{\frac {\left (-4 x^{2}+x \right ) \ln \left (3\right )-3 x^{3}}{4 \ln \left (3\right )+3 x}}+40 \,{\mathrm e}^{-2} \ln \left (3\right ) x \,{\mathrm e}^{\frac {\left (-4 x^{2}+x \right ) \ln \left (3\right )-3 x^{3}}{4 \ln \left (3\right )+3 x}}}{4 \ln \left (3\right )+3 x}\) | \(87\) |
int(((-320*x^2+40*x+160)*ln(3)^2+(-480*x^3+240*x)*ln(3)-180*x^4+90*x^2)*ex p(((-4*x^2+x)*ln(3)-3*x^3)/(4*ln(3)+3*x))/(16*exp(2)*ln(3)^2+24*x*exp(2)*l n(3)+9*x^2*exp(2)),x,method=_RETURNVERBOSE)
Time = 0.28 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.44 \[ \int \frac {e^{\frac {-3 x^3+\left (x-4 x^2\right ) \log (3)}{3 x+4 \log (3)}} \left (90 x^2-180 x^4+\left (240 x-480 x^3\right ) \log (3)+\left (160+40 x-320 x^2\right ) \log ^2(3)\right )}{9 e^2 x^2+24 e^2 x \log (3)+16 e^2 \log ^2(3)} \, dx=10 \, x e^{\left (-\frac {3 \, x^{3} + {\left (4 \, x^{2} - x\right )} \log \left (3\right )}{3 \, x + 4 \, \log \left (3\right )} - 2\right )} \]
integrate(((-320*x^2+40*x+160)*log(3)^2+(-480*x^3+240*x)*log(3)-180*x^4+90 *x^2)*exp(((-4*x^2+x)*log(3)-3*x^3)/(4*log(3)+3*x))/(16*exp(2)*log(3)^2+24 *x*exp(2)*log(3)+9*x^2*exp(2)),x, algorithm=\
Time = 0.20 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.24 \[ \int \frac {e^{\frac {-3 x^3+\left (x-4 x^2\right ) \log (3)}{3 x+4 \log (3)}} \left (90 x^2-180 x^4+\left (240 x-480 x^3\right ) \log (3)+\left (160+40 x-320 x^2\right ) \log ^2(3)\right )}{9 e^2 x^2+24 e^2 x \log (3)+16 e^2 \log ^2(3)} \, dx=\frac {10 x e^{\frac {- 3 x^{3} + \left (- 4 x^{2} + x\right ) \log {\left (3 \right )}}{3 x + 4 \log {\left (3 \right )}}}}{e^{2}} \]
integrate(((-320*x**2+40*x+160)*ln(3)**2+(-480*x**3+240*x)*ln(3)-180*x**4+ 90*x**2)*exp(((-4*x**2+x)*ln(3)-3*x**3)/(4*ln(3)+3*x))/(16*exp(2)*ln(3)**2 +24*x*exp(2)*ln(3)+9*x**2*exp(2)),x)
Time = 0.43 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.20 \[ \int \frac {e^{\frac {-3 x^3+\left (x-4 x^2\right ) \log (3)}{3 x+4 \log (3)}} \left (90 x^2-180 x^4+\left (240 x-480 x^3\right ) \log (3)+\left (160+40 x-320 x^2\right ) \log ^2(3)\right )}{9 e^2 x^2+24 e^2 x \log (3)+16 e^2 \log ^2(3)} \, dx=10 \cdot 3^{\frac {1}{3}} x e^{\left (-x^{2} - \frac {4 \, \log \left (3\right )^{2}}{3 \, {\left (3 \, x + 4 \, \log \left (3\right )\right )}} - 2\right )} \]
integrate(((-320*x^2+40*x+160)*log(3)^2+(-480*x^3+240*x)*log(3)-180*x^4+90 *x^2)*exp(((-4*x^2+x)*log(3)-3*x^3)/(4*log(3)+3*x))/(16*exp(2)*log(3)^2+24 *x*exp(2)*log(3)+9*x^2*exp(2)),x, algorithm=\
Time = 0.34 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.60 \[ \int \frac {e^{\frac {-3 x^3+\left (x-4 x^2\right ) \log (3)}{3 x+4 \log (3)}} \left (90 x^2-180 x^4+\left (240 x-480 x^3\right ) \log (3)+\left (160+40 x-320 x^2\right ) \log ^2(3)\right )}{9 e^2 x^2+24 e^2 x \log (3)+16 e^2 \log ^2(3)} \, dx=10 \cdot 3^{\frac {1}{3}} x e^{\left (-\frac {9 \, x^{3} + 12 \, x^{2} \log \left (3\right ) + 4 \, \log \left (3\right )^{2}}{3 \, {\left (3 \, x + 4 \, \log \left (3\right )\right )}} - 2\right )} \]
integrate(((-320*x^2+40*x+160)*log(3)^2+(-480*x^3+240*x)*log(3)-180*x^4+90 *x^2)*exp(((-4*x^2+x)*log(3)-3*x^3)/(4*log(3)+3*x))/(16*exp(2)*log(3)^2+24 *x*exp(2)*log(3)+9*x^2*exp(2)),x, algorithm=\
Time = 13.60 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.48 \[ \int \frac {e^{\frac {-3 x^3+\left (x-4 x^2\right ) \log (3)}{3 x+4 \log (3)}} \left (90 x^2-180 x^4+\left (240 x-480 x^3\right ) \log (3)+\left (160+40 x-320 x^2\right ) \log ^2(3)\right )}{9 e^2 x^2+24 e^2 x \log (3)+16 e^2 \log ^2(3)} \, dx=10\,3^{\frac {x-4\,x^2}{3\,x+\ln \left (81\right )}}\,x\,{\mathrm {e}}^{-\frac {3\,x^3}{3\,x+\ln \left (81\right )}-2} \]